Question

Area (in sq units) of the triangle formed by the lines $y=2x,y=3x$ and $y=5$ is

• A. $\frac{25}{6}$
• B. $\frac{25}{12}$
• C. $\frac{5}{6}$
• D. $\frac{17}{12}$
• E. $6$

Answer 1

The correct option is B

$\frac{25}{12}$

Explanation for the correct option:

Given that $y=2x,y=3x$ and $y=5$ are the equation of lines

The intersection point of these lines are $(0,0),\left(\frac{5}{2},5\right)$ and $\left(\frac{5}{3},5\right)$

$\therefore$The area of triangles $=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 5& 1\\ \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.& 5& 1\end{array}\right|$

$$\begin{gathered} =\frac{1}{2}\left[0-0+1\left(\frac{5}{2}\times 5\right)-\left(\frac{5}{3}\times 5\right)\right] \\ =\frac{1}{2}\left(\frac{25}{2}-\frac{25}{3}\right) \\ =\frac{1}{2}\left(\frac{75-50}{6}\right) \\ =\frac{1}{2}\times \frac{25}{6} \\ =\frac{25}{12} \end{gathered}$$

Thus, The area of the triangles is $\frac{\mathbf{25}}{\mathbf{12}}$ sq units

Hence, Option ‘B’ is Correct.